Important equivalence transformations for inequalities.
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Inequalities are written using the comparison signs \(<\), \(\leq\), \(>\), \(\geq\).
Important equivalence transformations:
addition/subtraction: \(\qquad a < b
\quad \Longleftrightarrow \quad a+c < b+c\)
multiplication with/division by a positive constant:
\[a < b \quad \Longleftrightarrow \quad
ac < bc \quad \Longleftrightarrow \quad \frac{a}{c} <
\frac{b}{c}\]
where \(c > 0\)
multiplication with/division by a negative constant and flipping
the comparison sign:
\[a < b \quad \Longleftrightarrow \quad
ac > bc \quad \Longleftrightarrow \quad \frac{a}{c} >
\frac{b}{c}\]
where \(c < 0\).
interchanging sides and flipping the comparison sign:
\[a < b \quad \Longleftrightarrow
\quad b > a\]
taking positive powers: \(\quad a <
b \quad \Longleftrightarrow \quad a^p < b^p\)
where \(a,b,p>0\).
The above transformations are also valid using the other comparison
signs.
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